Enumerating fuzzy switching functions and free kleene algebras

Normal forms and truth tables for fuzzy logics

“…See, for example, [2,3,4,5,8,10,11,14,15,16]. However, some of this mathematics does not seem to be well known within the fuzzy community.…”

Upper and lower bounds on the number of fuzzy/c switching functions

“…First, there are p possibilities to choose a constant with respect to each element of the set composed of s elements in V k . Secondly, for the above each case, considering all possibilities included the case that do not choose, there are p+1 possibilities to form a subset ( that is, an anti-chain with constants ) with respect to each element of the set which has the remaining |V | The above gives a lower bound on the number of n-variable fuzzy/c switching functions | | FC , specially in the case of p=1, it gives a lower bound that of n-variable fuzzy switching function | | F [5] [10] . So, transforming the above in consideration of a sign of inequality, it becomes as follows.…”

“…As generalization of a fuzzy switching function, a fuzzy switching function with arbitrary constants ( for short, a fuzzy/c switching function) whose constants are allowed to take an arbitrary value in the closed interval [ , ] 0 1 is proposed and studied by the authors [2] . Meanwhile, the enumerating problem of nvariable fuzzy switching functions has been studied by many researchers [1], [3], [4], [5] . This problem is essentially identical to enumerating elements in a certain kind of distributive lattice called the free Kleene algebra on n generators [5] , and it is known that enumerating them exactly is very difficult.…”

“…Meanwhile, the enumerating problem of nvariable fuzzy switching functions has been studied by many researchers [1], [3], [4], [5] . This problem is essentially identical to enumerating elements in a certain kind of distributive lattice called the free Kleene algebra on n generators [5] , and it is known that enumerating them exactly is very difficult. In fact, despite many attempts, the exact number has been reported only in the case of n=1, 2, 3, 4 as the number of variables [4], [5] .…”

“…This problem is essentially identical to enumerating elements in a certain kind of distributive lattice called the free Kleene algebra on n generators [5] , and it is known that enumerating them exactly is very difficult. In fact, despite many attempts, the exact number has been reported only in the case of n=1, 2, 3, 4 as the number of variables [4], [5] . On the other hand, the number of all n-variable fuzzy/c switching functions is infinite.…”